Initial value problem

As initial value problem (abbreviated AWP ), sometimes initial value problem (abbreviated AWA) or Cauchy problem mentioned is called in the analysis an important class of differential equations . The solution of an initial value problem is the solution of the differential equation with additional consideration of a given initial value .

In this article the initial value problem is explained first for ordinary differential equations and later also for partial differential equations .

Ordinary differential equations

1st order initial value problem

A first order initial value problem is a system of equations that consists of an ordinary first order differential equation

${\displaystyle y'(t)=f(t,y(t))}$

and an additional initial condition

${\displaystyle y(t_{0})=y_{0}}$

exists, with

• the initial value and ${\displaystyle y_{0}}$
• a point in time . ${\displaystyle t_{0}\in \mathbb {R} }$

A concrete function is a solution to the initial value problem if it satisfies both equations. ${\displaystyle y}$

We are looking for a function that fulfills the conditions of the differential equation and the initial value. If the function is continuous, then according to the main theorem of integral calculus this is the case if and only if ${\displaystyle y}$ ${\displaystyle f}$

${\displaystyle y(t)=y_{0}+\int _{t_{0}}^{t}f(s,y(s))\,ds}$

applies to all in the definition interval. ${\displaystyle t}$

K th order initial value problem

Given and a function . Let your domain of definition be a subset of , where denotes an interval that includes. Then is called ${\displaystyle k\in \mathbb {N} }$ ${\displaystyle f\colon D\rightarrow \mathbb {R} ^{n}}$ ${\displaystyle D}$ ${\displaystyle I\times \mathbb {R} ^{n\times k}}$ ${\displaystyle I\subset \mathbb {R} }$ ${\displaystyle t_{0}}$

${\displaystyle {\begin{cases}y^{(k)}&=f(t,y(t),y'(t),\dotsc ,y^{(k-1)}(t))\\y^{(i)}(t_{0})&=y_{i}\qquad \mathrm {mit} \;i=0,\dotsc ,k-1\end{cases}}}$

a -th order initial value problem . Every -th order initial value problem can be rewritten as a first-order initial value problem. ${\displaystyle k}$ ${\displaystyle k}$

A special initial value problem is the Riemann problem , in which the initial data are constant except for one point of discontinuity .

Initial value problems occur e.g. B. in the natural sciences when a mathematical model for natural processes is sought.

Solvability

Important records that affect the solubility of initial value problems for ordinary differential equations, the (local) existence set of Peano and the existence and uniqueness theorem of Picard-Lindelöf . One aid is the Grönwall inequality .

example

The initial value problem

${\displaystyle y'(t)=2\cdot {\rm {sgn}}(y(t))\cdot {\sqrt {|y(t)|}}\ ,\ y(0)=0\ ,}$

which to

${\displaystyle f(t,x):=2\cdot {\rm {sgn}}(x)\cdot {\sqrt {|x|}}}$

corresponds, has an infinite number of solutions, namely in addition to the trivial solution

${\displaystyle y(t)\equiv 0}$

also the solutions for each ${\displaystyle c\geq 0}$

${\displaystyle y(t)=\left\{{\begin{array}{ll}0\ ,&{\textrm {falls}}\ t<c\ ,\\(t-c)^{2}\ ,&{\textrm {falls}}\ t\geq c\ ,\\\end{array}}\right.}$

such as

${\displaystyle y(t)=\left\{{\begin{array}{ll}0\ ,&{\textrm {falls}}\ t<c\ ,\\-(t-c)^{2}\ ,&{\textrm {falls}}\ t\geq c\ .\\\end{array}}\right.}$

Additional properties (an ) must be demonstrated so that initial value problems have unambiguous solutions . This can be done, for example, using Picard-Lindelöf's theorem , whose prerequisites are not met in this example. ${\displaystyle f}$

Numerical solution methods

One-step or multi-step methods are used for the numerical solution of initial value problems . The differential equation is approximated by means of a discretization .

Partial differential equations

If one generalizes the Cauchy problem to several variables, such as variables , one obtains partial differential equations . The following stands for a multi-index of length . Note that there are exactly multi-indices with . Let there be a function in variables. The general Cauchy problem looks for functions that depend on variables and the equation ${\displaystyle n}$ ${\displaystyle x_{1},\dotsc ,x_{n}}$ ${\displaystyle \alpha \in \mathbb {N} _{0}^{n}}$ ${\displaystyle n}$ ${\displaystyle {\tbinom {n+k-1}{k}}}$ ${\displaystyle |\alpha |:=\alpha _{1}+\dotsb +\alpha _{n}\leq k}$ ${\displaystyle F}$ ${\displaystyle n+{\tbinom {n+k-1}{k}}}$ ${\displaystyle u}$ ${\displaystyle n}$ ${\displaystyle x_{1},\dotsc ,x_{n}}$

(1) ${\displaystyle F(x,(\partial ^{\alpha }u(x))_{|\alpha |\leq k)})\,=\,0}$

fulfill. Note that the arity of was chosen so that one and all partial derivatives can be used. In addition, it is required that the functions sought satisfy the so-called initial or boundary conditions described below. To formulate it, let a hypersurface of class C ^k with normal field . With which are normal derivatives designated. If functions are then given and defined, the general Cauchy problem requires that the functions also meet the conditions ${\displaystyle F}$ ${\displaystyle x=(x_{1},\dotsc ,x_{n})}$ ${\displaystyle \partial ^{\alpha }u(x)}$ ${\displaystyle S}$ ${\displaystyle \nu }$ ${\displaystyle \partial _{\nu }^{j}}$ ${\displaystyle \varphi _{0},\dotsc ,\varphi _{k-1}}$ ${\displaystyle S}$ ${\displaystyle u}$

(2) on ${\displaystyle u=\varphi _{0},\,\partial _{\nu }u=\varphi _{1},\,\dotsc ,\,\partial _{\nu }^{k-1}u=\varphi _{k-1}}$ ${\displaystyle S}$

fulfill. The functions are called the Cauchy data of the problem, every function that satisfies both conditions (1) and (2) is called a solution to the Cauchy problem. ${\displaystyle \varphi _{j}}$ ${\displaystyle u}$

A suitable coordinate transformation can be used to withdraw to the case . Then the last variable plays a special role, because the initial conditions are given where this variable is 0. Since this variable is interpreted as time in many applications, it is often renamed (Latin tempus = time), the initial conditions then describe the relationships at the point in time . So the variables are . Since the observed hyperplane is given by the condition , the normal derivative simply becomes the derivative according to . If one writes abbreviated and , then the Cauchy problem is now ${\displaystyle S=\{x=(x_{1},\dotsc ,x_{n});\,x_{n}=0\}}$ ${\displaystyle t}$ ${\displaystyle t=0}$ ${\displaystyle x_{1},\dotsc ,x_{n-1},t}$ ${\displaystyle t=0}$ ${\displaystyle t}$ ${\displaystyle x=(x_{1},\dotsc ,x_{n-1})}$ ${\displaystyle \alpha =(\alpha _{1},\dotsc ,\alpha _{n-1})}$

(1') ${\displaystyle F(x,t,(\partial _{x}^{\alpha }\partial _{t}^{j}u(x,t))_{|\alpha |+j\leq k)})\,=\,0}$

(2 ') . ${\displaystyle u(x,0)=\varphi _{0}(x),\,\partial _{t}u(x,0)=\varphi _{1}(x),\,\dotsc ,\,\partial _{t}^{k-1}u(x,0)=\varphi _{k-1}(x)}$

A typical example is the three-dimensional wave equation

${\displaystyle \partial _{t}^{2}u-c^{2}\cdot \Delta u=f}$

${\displaystyle u(x,0)=\varphi _{0}(x),\,\partial _{t}u(x,0)=\varphi _{1}(x)}$,

where be a constant, a given function and the Laplace operator . ${\displaystyle c}$ ${\displaystyle f}$ ${\displaystyle \Delta =\partial _{x_{1}}^{2}+\partial _{x_{2}}^{2}+\partial _{x_{3}}^{2}}$

Is a solution, which will imply the same time sufficient differentiation, all the derivatives are with already by the Cauchy data set, because it is . Only the derivation is not determined by (2 '), so only (1') can set a condition here. So that (1 ') is actually a non-trivial condition and so that the Cauchy problem is not badly posed from the start , one will require that equation (1') can be solved. The Cauchy problem then has the form ${\displaystyle u}$ ${\displaystyle \partial _{x}^{\alpha }\partial _{t}^{j}u(x,0)}$ ${\displaystyle |\alpha |+j\leq k,j<k}$ ${\displaystyle \partial _{x}^{\alpha }\partial _{t}^{j}u(x,0)=\partial _{x}^{\alpha }\varphi _{j}}$ ${\displaystyle \partial _{t}^{k}u}$ ${\displaystyle \partial _{t}^{k}u}$

(1") ${\displaystyle \partial _{t}^{k}u(x,t)=G(x,t,(\partial _{x}^{\alpha }\partial _{t}^{j}u(x,t))_{|\alpha |+j\leq k,j<k})}$

(2 ") , ${\displaystyle u(x,0)=\varphi _{0}(x),\,\partial _{t}u(x,0)=\varphi _{1}(x),\,\dotsc ,\,\partial _{t}^{k-1}u(x,0)=\varphi _{k-1}(x)}$

where is a suitable function of arity . In the formulation given last, all occurring derivatives have an order , and the -th derivative after actually occurs, because this is just the left side of (1 ") and it does not appear on the right side of (1"). The order of the Cauchy problem is therefore also called. The above example of the three-dimensional wave equation is obviously easy to put into this form, ${\displaystyle G}$ ${\displaystyle n-1+{\tbinom {n+k-1}{k}}}$ ${\displaystyle \leq k}$ ${\displaystyle k}$ ${\displaystyle t}$ ${\displaystyle k}$

${\displaystyle \partial _{t}^{2}u=f+c^{2}\cdot \Delta u}$

${\displaystyle f}$ ${\displaystyle u(x,0)=\varphi _{0}(x),\,\partial _{t}u(x,0)=\varphi _{1}(x)}$

there is therefore a Cauchy problem of order 2.

If all Cauchy data are analytical, the Cauchy-Kowalewskaja theorem ensures unambiguous solutions to the Cauchy problem.

Determination of the constant of integration

In school mathematics, the determination of the constant of integration of an indefinite integral for a given point is called the initial value problem.

example

We are looking for the antiderivative of the fractional-rational function given by ${\displaystyle F_{C}}$ ${\displaystyle f}$

${\displaystyle f(x)={\frac {1}{x^{2}-2x+1}}}$,

that goes through the point . ${\displaystyle P(4|5)}$

First we factor the denominator:

${\displaystyle f(x)={\frac {1}{(x-1)^{2}}}=(x-1)^{-2}}$.

Now we can substitute :

${\displaystyle \int u^{-2}\cdot 1\,\mathrm {d} u=-u^{-1}=-{\frac {1}{x-1}}+C}$.

Next we need to plug in the x coordinate of the point and set the term equal to the y value

${\displaystyle -{\frac {1}{4-1}}+C=5\Leftrightarrow C={\frac {16}{3}}}$.

The antiderivative we are looking for is therefore:

${\displaystyle F_{C}(x)=-{\frac {1}{x-1}}+{\frac {16}{3}}}$.

Abstract Cauchy problem

Let be a Banach space and be a linear or nonlinear operator. The question of whether a given , and a differentiable function with all exist, the initial value problem the ${\displaystyle X}$ ${\displaystyle A\colon D(A)\subset X\rightarrow X}$ ${\displaystyle T>0}$ ${\displaystyle u_{0}\in X}$ ${\displaystyle f\colon (0,T)\rightarrow X}$ ${\displaystyle u\colon [0,T)\rightarrow X}$ ${\displaystyle u(t)\in D(A)}$ ${\displaystyle T>t>0}$

${\displaystyle {\begin{matrix}u'(t)+A(u(t))&=&f(t),&\quad T>t>0\\u(0)&=&u_{0}&\end{matrix}}}$

is called the abstract Cauchy problem . To be able to solve them, one needs the theory of the strongly continuous semigroups or the analytical semigroups . For the different initial conditions and operators there are different types of the solution concept, in the linear distributional solution, in the nonlinear the integral solution. The downstream regularity theory deals with classically differentiable or almost everywhere differentiable solutions.

literature

• Wolfgang Walter: Ordinary Differential Equations: An Introduction. 7th edition. Springer, 2000, ISBN 3-540-67642-2 .
• Isao Miyadera, Choong Yun Cho: Nonlinear Semigroups. American math. Soc., Providence, RI 1992, ISBN 0-8218-4565-9 .
• Amnon Pazy: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, New York 1983, ISBN 0-387-90845-5 .
• Gerald B. Folland: Introduction to Partial Differential Equations. Princeton University Press, 1976, ISBN 0-691-08177-8 . (especially Chapter 1.C. for the general Cauchy problem)

Web links

• Gert Lube: Initial value problems. (Script, University of Göttingen )
• Clemens Brand: Illustrations for a simple initial value problem. (Script, University of Leoben )
• taramath online tool for solving initial value problems.

Individual evidence

1. ^ Rannacher, Rolf: Numerics 1. Numerics of ordinary differential equations . Heidelberg 2017, p. 13 .
2. ^ Anton Bigalke, Norbert Köhler: Mathematics. Grammar school upper level Berlin basic course ma-2. Cornelsen Verlag / Volk und Wissen Verlag, Berlin 2011, ISBN 978-3-06-040002-7 , p. 27 .